 oldandrew:Not only do I not think that is true and if it was it would be pretty useless in exams, but it is still slower than doing it in your head. You could always have put the quadratic formula in a programmable calculator if you wanted to, even back when I was at school. However, it is no substitute for mathematical fluency. The best A-level maths students can factorise that sort of question in their head in less time than it takes to say the answer. The Chinese system seems to be producing a lot more of that sort of student than ours does. The scary thing, though, is that people are making excuses for that dumbing down and even pretending there is something wrong with the Chinese system for not having gone the same way.
No-one can factorise integers in better than O(sqrt(N)) time. Which means that almost no-one can do it at all unless the integers are pretty small. I'm not aware of any sphere of applied mathematics where small quadratics crop up very frequently. That's not to say there are no such spheres, but most people, even if they go on to use mathematics professionally, won't encounter them. The reason for teaching quadratics at elementary levels is to develop mathematical fluency - get pupils used to algebraic manipulations, brackets, converting between algebraic and graph representations, and, admittedly, also to develop mental arithmetic. Exams are a means to an end, not an end in themselves, even if it doesn't seem like it with the current regime. However I'm principally interested in mathematical fluency with a computer. Everyone now has access to cheap computers - I'm typing this at one now. The ability to do mental arithmetic isn't of much use to me, the computer does three billion calculations a second, impossible for any human to even approach, much less to beat. Other people have made the point that degree-level mathematics, as opposed to what I do (statistics and mathematical modelling), is done without electronic assistance, which is maybe a legitimate reason for an emphasis on mental arithmetic in the early stages. I don't pretend to have all the answers, but we do need a mature debate on how to respond to the advent of computers, and what that means for maths education.
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